Poyet revealed on Monday that the former Bolton Wanderers left-back will return to the Premier League on a six-month deal when the transfer window opens on Wednesday after failing to establish himself in the Fiorentina starting line-up.
"I can confirm Marcos is going to be with us for the rest of the season," said the Uruguayan coach.
"I was looking for a player in that position on the left side and I hope he will help us and everybody will like him.
"I am delighted that everything is agreed."
Alonso joined the Florence club from Bolton in May, but has made only three Serie A appearances for Vincenzo Montella's side.
Sunderland are bottom of the Premier League, but the Wearside outfit are unbeaten in their last five games in all competitions ahead of their crunch clash with fellow strugglers Aston Villa on New Year's Day.
